We study the pair correlation function for a variety of completely integrable quantum maps in one degree of freedom. For simplicity we assume that the classical phase space M is the Riemann sphere ℂP1 and that the classical map is a fixed-time map exptΞH of a Hamilton flow. The quantization is then a unitary N x N matrix Ut,N, and its pair correlation measure ρ(N)2, gives the distribution of spacings between eigenvalues in an interval of length comparable to the mean level spacing (∼ 1/N). The physicists' conjecture (Berry-Tabor conjecture) is that as N → ∞, ρ(N)2,t should converge to the pair correlation function ρPOISSON2 = δo + 1 of a Poisson process. For any 2-parameter family of Hamiltonians of the form Hα,β = αφ(Î) + βÎ with φ″ ≠ 0 we prove that this conjecture is correct for almost all (α, β) along the subsequence of Planck constants Nm = [m(log m)5]. In the addendum to this paper [Z. Addendum], we further show that for polynomial phases φ the a.e. convergence to Poisson holds along the full sequence of Planck constants for the Cesaro means of ρ(N)2;(t,α,β).
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics